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Electromagnetism 

In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.^{[1]}^{[2]}
Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector P.^{[2]}
An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements. These are elements which are bound to molecules and are not free to move around the material. Positive charged elements are displaced in the direction of the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms.^{[3]}^{[4]}
For a certain volume element in the material, which carries a dipole moment , we define the polarization density P:
In general, the dipole moment changes from point to point within the dielectric. Hence, the polarization density P of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole moment dp is:

(1) 
The net charge appearing as a result of polarization is called bound charge and denoted .
This definition of polarization density as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes.^{[5]}
Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge will be displaced a distance relative to the negative bound charge , giving rise to a dipole moment . Substitution of this expression in (1) yields
Since the charge bounded in the volume dV is equal to the equation for P becomes:^{[3]}

(2) 
where is the density of the bound charge in the volume under consideration. It is clear from the definition above that the dipoles are overall neutral, that is balanced by an equal density of the opposite charge within the volume. Charges that are not balanced are part of the free charge discussed below.
For a given volume V enclosed by a surface S, the bound charge inside it is equal to the flux of P through S taken with the negative sign, or

(3) 
Let a surface area S envelope part of a dielectric. Upon polarization negative and positive bound charges will be displaced. Let d_{1} and d_{2} be the distances of the bound charges and , respectively, from the plane formed by the element of area dA after the polarization. And let dV_{1} and dV_{2} be the volumes enclosed below and above the area dA.
It follows that the negative bound charge moved from the outer part of the surface dA inwards, while the positive bound charge moved from the inner part of the surface outwards.
By the law of conservation of charge the total bound charge left inside the volume after polarization is:
Since
The above equation becomes
By (2) it follows that , so we get:
And by integrating this equation over the entire closed surface S we find that
which completes the proof.
By the divergence theorem, Gauss's law for the field P can be stated in differential form as:
By the divergence theorem we have that
In a homogeneous, linear, nondispersive and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E:^{[7]}
where ε_{0} is the electric constant, and χ is the electric susceptibility of the medium. Note that in this case χ simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the isotropy of the dielectric.
Taking into account this relation between P and E, equation (3) becomes:^{[3]}
The expression in the integral is Gauss's law for the field E which yields the total charge, both free and bound , in the volume V enclosed by S.^{[3]} Therefore,
which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume):
Since within a homogeneous dielectric there can be no free charges , by the last equation it follows that there is no bulk bound charge in the material . And since free charges can get as close to the dielectric as to its topmost surface, it follows that polarization only gives rise to surface bound charge density (denoted to avoid ambiguity with the volume bound charge density ).^{[3]}
may be related to P by the following equation:^{[8]}
The class of dielectrics where the polarization density and the electric field are not in the same direction are known as anisotropic materials.
In such materials, the ith component of the polarization is related to the jth component of the electric field according to:^{[7]}
This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field of crystal optics.
As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomicscale behaviors. The polarizability of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius–Mossotti relation.
In general, the susceptibility is a function of the frequency ω of the applied field. When the field is an arbitrary function of time t, the polarization is a convolution of the Fourier transform of χ(ω) with the E(t). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and causality considerations lead to the Kramers–Kronig relations.
If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:
where is the linear susceptibility, is the secondorder susceptibility (describing phenomena such as the Pockels effect, optical rectification and secondharmonic generation), and is the thirdorder susceptibility (describing thirdorder effects such as the Kerr effect and electric fieldinduced optical rectification).
In ferroelectric materials, there is no onetoone correspondence between P and E at all because of hysteresis.
The behavior of electric fields (E, D), magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations in matter.
In terms of volume charge densities, the free charge density is given by
where is the total charge density. By considering the relationship of each of the terms of the above equation to the divergence of their corresponding fields (of the electric displacement field D, E and P in that order), this can be written as:^{[9]}
This is known as the constitutive equation for electric fields. Here ε_{0} is the electric permittivity of empty space. In this equation, P is the (negative of the) field induced in the material when the "fixed" charges, the dipoles, shift in response to the total underlying field E, whereas D is the field due to the remaining charges, known as "free" charges.^{[5]}^{[10]}
In general, P varies as a function of E depending on the medium, as described later in the article. In many problems, it is more convenient to work with D and the free charges than with E and the total charge.^{[1]}
Therefore, a polarized medium, by way of Green's Theorem can be split into four components.
When the polarization density changes with time, the timedependent boundcharge density creates a polarization current density of
so that the total current density that enters Maxwell's equations is given by
where J_{f} is the freecharge current density, and the second term is the magnetization current density (also called the bound current density), a contribution from atomicscale magnetic dipoles (when they are present).
The polarization inside a solid is not, in general, uniquely defined. Because a bulk solid is periodic, one must choose a unit cell in which to compute the polarization (see figure).^{[11]}^{[12]} In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of P, and neither of them will be wrong. For example, if Alice chooses a unit cell with positive ions at the top and Bob chooses the unit cell with negative ions at the top, their computed P vectors will have opposite directions. Alice and Bob will agree on the microscopic electric field E in the solid, but disagree on the value of the displacement field .
On the other hand, even though the value of P is not uniquely defined in a bulk solid, variations in P are uniquely defined.^{[11]} If the crystal is gradually changed from one structure to another, there will be a current inside each unit cell, due to the motion of nuclei and electrons. This current results in a macroscopic transfer of charge from one side of the crystal to the other, and therefore it can be measured with an ammeter (like any other current) when wires are attached to the opposite sides of the crystal. The timeintegral of the current is proportional to the change in P. The current can be calculated in computer simulations (such as density functional theory); the formula for the integrated current turns out to be a type of Berry's phase.^{[11]}
The nonuniqueness of P is not problematic, because every measurable consequence of P is in fact a consequence of a continuous change in P.^{[11]} For example, when a material is put in an electric field E, which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift. This changes P, and the result is electric susceptibility (and hence permittivity). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing P. The result is pyroelectricity. In all cases, the properties of interest are associated with a change in P.
Even though the polarization is in principle nonunique, in practice it is often (not always) defined by convention in a specific, unique way. For example, in a perfectly centrosymmetric crystal, P is exactly zero due to symmetry reasoning.
This can be seen in a pyroelectric material. Above the Curie temperature the material is not polarized and it has a centrosymmetric configuration. Lowering the temperature below the Curie temperature induces a structural phase transition that breaks the centrosymmetricity. The P of the material grows proportionally to the distortion, thus allowing to define it unambiguously.
Another problem in the definition of P is related to the arbitrary choice of the "unit volume", or more precisely to the system's scale.^{[5]} For example, at microscopic scale a plasma can be regarded as a gas of free charges, thus P should be zero. On the contrary, at a macroscopic scale the same plasma can be described as a continuous medium, exhibiting a permittivity and thus a net polarization P ≠ 0.